weekly, and nioiUhly data from 1962 to 1994. Wc use market capitalization to group securities because the relative thinness of the market for any given stock is highly correlated with the stocks total market value; h-sticc slocks with similar market values are likely to have similar nontrading probabilities.39 We choose to form ten portfolios to maximize the homogeneity of nontrading probabilities within each portfolio while still maintaining reasonable diversification so that the asymptotic approximation of (il.20) might still obtain.40

Dily Nontrading Probabilities Implicit in Autocorrelations Table 3.4 reports first-order autocorrelation matrices Г\ for the vector of four of the ten size-sorted portfolio returns using daily, weekly, and monthly data taken from the Center for Research in Security Prices (CRSP) database. Pc rtfolio 1 contains stocks with the smallest market values and portfolio 10 contains those with the largest.41 From casual inspection il is apparent that these autocorrelation matrices arc not symmetric. The second column of matrices is the autocorrelation matrices minus their transposes, and it is evident that elements below the diagonal dominate those above it. This confirms the lead-lag pattern reported in Lo and MacKinlay (1990c).

j The fact that the returns of large stocks tend to lead those of smaller slocks docs suggest that nonsynchronous trading may be a source of correlation. However, the magnitudes of the autocorrelations for weekly and monthly returns imply an implausible level of nontrading. This is most evident in Table 3.5, which reports estimates of daily nontrading probabilities implicit in the weekly and monthly own-autocorrelations of Table 3.4.

For example, using (3.1.40) the daily nontrading probability implied by an estimated weekly autocorrelation of 37% for portfolio 1 is estimated to be 71.7%.42 Using (3.1.8) wc estimate the average time between trades lo

Only ordinary common shares are included in this analysis. Excluded are American Depository Receipts (ADRs) and oilier specialized securities where using market value to characterize nontrading is less meaningful.

4,The returns to these portfolios are continuously compounded returns of individual simple returns arithmetically averaged. We have repeated the correlation analysis lor continuously compounded returns of portfolios whose values are calculated as unweighted geometric averages of included securities prices. The results for these portfolio returns are practically identical to those for the continuously compounded returns ol equal-weighted portfolios.

41 We report only a subset of four portfolios for the sake of brevity.

42Slandard errors for autocorrelation-based probability and nontrading duration estimates are obtained by applying a lust-order Taylor expansion (see Section A.4 of the Appendix) lo (11.1.8) and (3.1.40) using heteroskedasticity- and autocorrelaiiotKonsistcnl standard errors for daily, weekly, and monthly first-order autocorrelation coefficients. These latter standard errors are computed by regressing returns on a constant and lagged returns, and using Newey ami Wests (1987) procedure to calculate heteroskedasticity- and autocorrelation-consistent standard errors for the slope coellicient (which is simply the lira-order autocorrelation coefficient of returns).

Table 3.4. Autocorrelation matrices for size-sorted portfolio returns.

Daily

Weekly

Monthly

Sample lirsi-ordei autocorrelation matrix Г lor the tlxl) suhvet lor i t{ rj r!j [ ) of observed returns lo ten equal-weighted si/.e-sorted portfolios using daily, weekly, and monthly NYSE-АМЫХ common stock ret tiros data from the CRSP hies lor the time period July 3, I *K 2 lo December 31), HUM. Stocks are assigned to portfolios annually using the market value at the end of the prior year. If this market value is missing the end ol year market value is used, It inith market values are missing the slock is not included. Only securities with complete daily reliun histories within л given month are included tn the daily returns calculations, ij is the return to the portfolio containing securities with the smallest mat kct values and rjn is the return to the portJolio of securities with the largest. There are approximately equal numbers of .securities in each portfolio. The entry in the ilh row and jth column is (he correlation between rt and ; ч,-To gauge the degree of asymmetry in these autocon elation matrices, the difference V\ - Г, is also reported.

be 2.5 days! The corresponding daily nontrading probability is 80.0% using monthly returns, implying an average nontrading duration of 0.5 days.

For comparison Tabic 3.5 also reports estimates al the nontrading prol>-abilitics usingdailydata and using trade information from the CRSP files. In the absence of lime aggregation own .mux onelations of portfolio returns are consistent estimators of nontrading probabilities; thus ihe entries in tbe column of Table 3.5 labelled nK(q = 1) are simply taken from the diagonal of the autocovariance matrix in Table 3,4.

For the smaller securities, the point estimates yield plausible nontrading durations, but the estimated durations decline only marginally lor larger-

Table 3.5. Estimates of ilailt nontrading probabilities.

(.Miniates of daily unutraditig probabilities implicit in ton weekly and monthly si/.e-sortcd pott-(oliuretuiuauiticui relations, funics in the column labelled Л, me averagesut the fraction of securities in poitliitin ih.ti iliil mil Hade tin each trading day, where the average is computed mrr all uading ilavslroni July 3. 10112 in Uei ember 30,1994. Kniriesin the )!. (,/ = 1) column are the IusHiiilet aiiKieiiiielaliini coefficients tif daily portfolio returns, which are ronsistent estimators iitdaily unnirading probabilities, ratifies in ilie я4(/ = 5) and п (1/ = 22) columns are estimates ol daily nnnirading probabilities ohl.lined (nun lirsi-tudrr weekly and liKiulldy porllotio return auliiroi relation eoellieienls, using the linie aggregation relations ol Scclion 3.2 (</ = 5 lor weekly leiiirns and </ = 22 lor monthly returns since there are 5 and 22 trading days in a week and a mouth, respec lively). Knit its in columns labelled A, are esti-ntaies ol ilie expected number ol conseculive days without nailing implied hy the probability csiiin.ilcs in columns to the iiimiediaie left. Standard errors are reported in parentheses; all are helcroskfdasiieity- and anionirrclalion-cnusisicut.

size portfolios. Л dm alion of nearly one fourth of a day is much too large for securities in the largest portfolio. More direct evidence is provided in the column labelled 7rk, which reports the average fraction of securities in a given portfolio that do not trade during each trading day.11 This average is computed over all trading days from July .3, 1002 lo December 30, 1994 (8179 observations). Comparing the entries in this column with those in the others shows the limitations of nontrading as an explanation for the autot orrclalions in the data. Nontrading may be responsible for some of the time-series properties of stock returns but cannot be the only source of autocorrelation.

I Ins mini malum is pi oviilcd in the (.KSI daily lilesin which llir closing price ol a security is ic polled lo be ilie negaiite ol the average ot the bid and ask prices on days when thai set airily did not nade. Sialid.ud errors lor probability estimates are based on ilie daily time series ol Ilie Iraciiiin ol noirades. Ilie standard mors are hetfroskedasticity- and auiocorrelaiion-i onsistent.

Implied lust-order autocorrelation p\ of weekly returns of an equal-weighted portfolio often .size-sorted portfolios (which approximates an equal-weighted portfolio of all securities), using two dillereiit estimators ofdaily nontrading probabilities for the portfolios: the average fraction of negative share prices reported by С RSI, and daily nonlrading probabilities implied by t rst-ordet autocorrelations ofdaily returns. Since the index autocorrelation depends on the betas of tin- ten portfolios, il is computed for two sets of betas, one in which all betas are set to 1.0 anil another in which the betas decline linearly from fl\ = 1.5 to fiM = 0.5. The sample weekly autocorrelation for an equal-weighted portfolio of the ten portfolios is 0.21. Results are basetl on data from July 3, 1902 to December 3(1, 1УУ4.

Nonsynchronous Trading and Index Autocorrelation Denote by r°, the observed return in period I to an equal-weighted portfojio of all N securities. Its autocovariance and autocorrelation are readily shown

to he I

CovK, C+ ] = J. СоггГС C+ ] = , (3.4.jl)

where Г is the contemporaneous covariance matrix of r° and i is an (Nx 1) vector of ones. If the betas of the securities are generally of the same sign and if the mean return of each security is small, then r°, is likely to be positively autocorrelated. Alternatively, if the cross-autocovariances are positive and dominate the negative own-autocovariances, the equal-weighted index will exhibit positive serial dependence. Can this explain Lo and MacKinlays (1988b) strong rejection of the random walk hypothesis for the CRSP weekly equal-weighted index, which exhibits a first-orderautocorrelation over 20%?

With little loss in generality we let N = 10 and consider the equal-weighted portfolio of the ten size-sorted portfolios, which is an approximately equal-weighted portfolio of all securities. Using (3.1.36) we may calculate the weekly autocorrelation of r°nl induced by particular daily non-trading probabilities тг,- and beta coefficients /},. To do this, we need to

select empirically plausible values for тг, and /) i = 1,2.....10. This is

done in Table 3.6 using two different methods of estimating the 7T;s and two different assumptions for the /J,s.

The first row corresponds to weekly autocorrelations computed with the nontrading probabilities obtained from the fractions of negative share prices reported by CRSP (see Table 3.5). The first entry, 1.4%, is the first-

Table 3.6. Nontrading-implied weekly index autocorreUttions.

order autocorrelation of the weekly equal-weighted index assuming that all t venty portfolio betas arc 1.0, and the second entry, 1.8%, is computed under the alternative assumption that the betas decline linearly from ftt = 1.5 for the portfolio of smallest stocks to pin = 0.5 for the portfolio of the hrgest. The second row reports similar autocorrelations implied by non-tiading probabilities estimated from daily autocorrelations using (3.1.41).

The largest implied first-order autocorrelation for the weekly equally lighted returns index reported in Table 3.6 is only 5.9%. Using direct e; timates of nontrading via negative share prices yields an autocorrelation til less than 2%. These magnitudes are still considerably smaller than the 21% sample autocorrelation of the equal-weighted index return. In summary, the recent empirical evidence provides little support for nontrading as an injiportant source of spurious correlation in the returns of common stock over daily and longer frequencies.44

3.4.2 Estimating the Effective Bid-Ask Spread

In implementing the model of Section 3.2.1, Roll (1984) argues that the percentage bid-ask spread sT may be more easily interpreted than the al>-solute bid-ask spread s, and he shows that the first-order autocovai iance of simple returns is related to sr in the following way:

Cov[ /t,-

Hi 1 =

Sr =

-~- (3.4.2)

(3.4.3)

where sr is defined as a percentage of the geometric average of the average bid and ask prices P and Pb. Using the approximation in (3.4.2), the percentage spread may be recovered as

2v/-Cov[/J, 1,/{,]

(3.4.4)

Note that (3.4.4) and (3.2.9) are only welklefincd when the return auio-covariance is negative, since by construction the bid-ask bounce can only induce negative first-order serial correlation. However, in practice, positive serial correlation in returns is not uncommon, and in these cases. Roll simply defines the spread to be (see footnotes a and b of his Table 1):

(3,1.5)

Boudotikh, Richardson, and Whitclaw (<>УГ>), Mccli (l.).)3) and Sias and Stalks (I.l.M) present additional empirical results on nontrading as a source of autocorrelation. While the papers do not agree on the level of autocorrelation induced by nontrading, all three papers conjlude that nontrading cannot completely account for the observed autocoi relations.

This convention seems difficult to justify on economic grounds-negative spreads are typically associated with markctmaking activity, i.e., Ihe provision of liquidity, yet this seems to have little connection with the presence of positive serial correlation in returns. Л more plausible alternative interpretation of cases where (3.4.4) is complex-valued is (hat the Roll (1984) model is misspccilicd ami that additional structure must be imposed to account for the positive serial correlation (see, for example, George el al. [1991], Gloslen and Harris [1988], Huang and Stoll [1995a], and Stoll [1989]).

Roll estimates the effective spreads of NYSE ami AMEX slocks year by year using daily returns data from 1963 lo 1982, and finds the overall average effective spread to be 0.298% for NYSE stocks and 1.74% for ЛМЕХ slocks (recall thai ЛМЕХ stocks tend lo be lower-priced; hence they ought to have larger percentage spreads). However, these figures must be interpreted with caution since 24,358 of the 47,414 estimated effective spreads were negative, suggesling the presence of substantial specification errors. Perhaps another symptom of diesc specification errors is die fact that estimates of the effective spread based on weekly data differ significantly from those based on daily data. Nevertheless, the magnitudes of these effects arc clearly important for empirical applications of transactions data.

Gloslen and Harris (1988) refine and estimate Glostcns (1987) decomposition of the bid-ask spread using transactions data for 250 NYSE slocks and conclude that the permanent adverse-selection component is indeed present in the data. Stoll (1989) develops a similar decomposition of the spread, and using transactions data for National Market System securities on the NASDAQ system from October to December of 1984, he concludes lhat 43% of the quoted spread is due to adverse selection, 10% is due to inventory-holding costs, and die remaining 47% is due lo order-processing costs. George, Kaul, and Nimalcndran (1991) allow the expected return of ihe unobscrvable true price (/* in the notation of Section 3.2.1) lo vary through lime, and using daily and weekly data for NYSE and ЛМЕХ stocks from 1963 to 1985 and NASDAQstocksand from 1983 to 1987, they obtain a much smaller estimate for the portion of the spread attributable to adverse selection-8% to 13%-with the remainder due lo order-processing costs, and no evidence of inventory costs. Huang and Stoll (1995a) propose a more general model that contains these other specifications as special cases and estimate the components of the spread to be 21% adverse-selection costs, 14% inventory-holding costs, and (>5% order-processing costs using 1992 transactions data for 19 of the 20 stocks in the Major Market Index.

The lacl that these estimates vary so much across studies makes il tlil-ficult lo regard any single study as conclusive. The differences come from two sources: different specifications for the dynamics of the bid-ask spread, and the use of different dalasets. There is clearly a need for a more detailed and comprehensive analysis in which all of these specifications are applied